Unified microscopic theory of equilibrium thermodynamics and ion association in aqueous and non-aqueous electrolytes with explicit hard-core size

Abstract

Within the framework of a functional integral formalism incorporating ionic charge and hard-core (HC) interactions on an equal footing, we formulate a unified theory of equilibrium thermodynamics and ion association in charged solutions. Via comparison with recent Monte-Carlo (MC) simulation results (J. Forsman et al., PCCP 26, 19921 (2024)), it is shown that our approach is able to predict with quantitative precision the pair distributions of monovalent ions with the typical hydrated sizes d = 3.0 A and 4.0 A up to the molar concentration ni = 2.0 M. Moreover, comparison with additional simulation data from the literature indicates that within the characteristic regime of ionic packing fraction eta <0.1, the formalism can accurately account for the ion size dependence of the excess energy and pressure from d = 14.3 A down to d = 1.6 A. Via the adjustment of the hydration radius, the theory can also reproduce the non-monotonic salt dependence of the experimentally measured osmotic coefficients of various aqueous and non-aqueous solutions. In accordance with AFM experiments involving non-aqueous electrolytes, the underlying sharp competition between the opposite charge attraction and the excluded volume constraint is shown to limit the occurrence of substantial ionic pair formation to the submolar concentration regime ni <50 mM; at larger concentrations, HC repulsion hinders ion association and results in the quasi-saturation of the pair fraction curves.

Figures (11)

• Figure 1: (Color online) Cation-anion (left plots) and cation-cation (right plots) pair distributions at two HC sizes and various monovalent salt concentrations ($q_i=\pm1$). Solid curves: SCDH prediction from Eq. (\ref{['eq86']}) via Eq. (\ref{['eq42']}). Circles: MC data from Figs. 1 and 2 of the supplementary material of Ref. Forsman. Dotted curves: DH prediction $g_{ij}(r)=1-q_iq_j v_{\rm DH}(r)$ with $v_{\rm DH}(r)=\ell_{\rm B}e^{-\kappa_{\rm DH}r}/r$ for $d=3$ Å rem1. The temperature and dielectric permittivity are $T=298$ K and $\varepsilon_{\rm s}=78.3$.
• Figure 2: (Color online) (a)-(b) Two-point correlation functions for the HC sizes $d=3$ Å and $4$ Å. Solid curves: SCDH prediction (\ref{['eq63']}). Dashed curves: DH potential $v_{\rm DH}(r)$. (c) Screening length rescaled by the DH length versus the salt concentration and (d) against the rescaled DH screening parameter. Solid curves: SCDH prediction (\ref{['eq58']}). Symbols: MC and HNC data from Figs.11 (a)-(b) of Ref. Forsman. The temperature and dielectric permittivity of the monovalent electrolyte ($q_i=\pm1$) are $T=298$ K and $\varepsilon_{\rm s}=78.3$ in all figures.
• Figure 3: (Color online) (a) Pair distribution functions, and (b)-(d) ionic charge density $\rho_{\rm c}(r)=q_+n_+H_{-+}(r)+q_-n_-H_{--}(r)$ around a central anion in 1:1 solutions. In (a), the reduced ion size is $d/\ell_{\rm B}=1/3$, and the packing fraction is $\eta=\pi n_id^3/3=0.01$. The MC data are from Fig.1 of Ref. NetzMC. In (b)-(d), the packing fractions are respectively $\eta=0.01$, $0.03$, and $0.15$. The reduced ion size is $d/\ell_{\rm B}=2/9$. The MC data are from Fig. 8B of Ref. NetzMC. The physical ion sizes and concentrations corresponding to the temperature $T=298$ K and dielectric permittivity $\varepsilon_{\rm s}=78.5$ are indicated in the legends.
• Figure 4: (Color online) (a) Energy and (b) pressure against the monovalent ion concentration ($q_i=\pm1$) at the temperature $T=298$ K, the dielectric constant $\varepsilon_{\rm s}=78.5$, and the HC size $d=4.25$ Å. Solid curves: SCDH predictions from Eqs. (\ref{['eq89']})-(\ref{['eq90']}). Symbols: MC data from (a) Ref. Val80 and (b) Ref. Svensson.
• Figure 5: (Color online) (a) Energy and (b) pressure against packing fraction $\eta=\pi n_id^3/3$ (lower horizontal axis) and reduced ion density $\rho^*=2n_id^3$ (upper horizontal axis) at various ion sizes. Solid curves: SCDH predictions from Eqs. (\ref{['eq89']})-(\ref{['eq90']}). Disk symbols: MC data from (a) Table 1 and (b) Table 2 of Ref. NetzMC. Triangles in (a): MC data from Fig.4 of Ref. Val90. From top to bottom, the reduced ion sizes or equivalently temperatures are $d/(\ell_{\rm B}q_i^2)=T^*=2$, $2/3$, $1/3$, $2/9$, $1/6$, and $0.1471$, respectively. The corresponding physical ion sizes for $T=298$ K, $\varepsilon_{\rm s}=78.5$, and $q_i=\pm1$ are given in the legends. (c) Pair fraction $\alpha$ from Eq. (\ref{['eq92']}) on linear and (d) semi-logarithmic scales at the parameters of the upper plots.